M. G. Krein’s lectures on entire operators. (English) Zbl 0883.47008

Gorbachuk, M. L.; Gorbachuk, V. I.

Operator Theory: Advances and Applications. 97. Basel: Birkhäuser. ix, 220 p. (1997).

The book deals with the results of M. G. Krein (and others) on so-called entire operators and their applications. During many years these results were mostly announced, not published in full details. The detailed presentation is contained in unpublished Krein’s lectures which are quite unaccessable for most of readers. Therefore, the work by M. L. Gorbachuk and V. I. Gorbachuk overcomes an essential gap in mathematical literature.
The main body of the book is devoted to entire operators with the deficiency index \((1.1)\) and their applications to various classical problems of analysis, like the power moment problem. This is the original subject of Krein’s works. Further development of the theory is reflected in three appendicies. The book is very rich, but self-contained. It is clearly written and may be recommended to anybody who wishes to learn this remarkable branch of classical and functional analysis.

Reviewer: A.Pankov (Vinnitsa)

MSC:

47A57	Linear operator methods in interpolation, moment and extension problems
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
47B99	Special classes of linear operators
35P99	Spectral theory and eigenvalue problems for partial differential equations

Keywords:

entire operators; deficiency index; power moment problem